Shot noise

Last updated
Photon noise simulation. Number of photons per pixel increases from left to right and from upper row to bottom row. Photon-noise.jpg
Photon noise simulation. Number of photons per pixel increases from left to right and from upper row to bottom row.

Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process.

Contents

In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where shot noise is associated with the particle nature of light.

Origin

In a statistical experiment such as tossing a fair coin and counting the occurrences of heads and tails, the numbers of heads and tails after many throws will differ by only a tiny percentage, while after only a few throws outcomes with a significant excess of heads over tails or vice versa are common; if an experiment with a few throws is repeated over and over, the outcomes will fluctuate a lot. From the law of large numbers, one can show that the relative fluctuations reduce as the reciprocal square root of the number of throws, a result valid for all statistical fluctuations, including shot noise.

Shot noise exists because phenomena such as light and electric current consist of the movement of discrete (also called "quantized") 'packets'. Consider lighta stream of discrete photonscoming out of a laser pointer and hitting a wall to create a visible spot. The fundamental physical processes that govern light emission are such that these photons are emitted from the laser at random times; but the many billions of photons needed to create a spot are so many that the brightness, the number of photons per unit of time, varies only infinitesimally with time. However, if the laser brightness is reduced until only a handful of photons hit the wall every second, the relative fluctuations in number of photons, i.e., brightness, will be significant, just as when tossing a coin a few times. These fluctuations are shot noise.

The concept of shot noise was first introduced in 1918 by Walter Schottky who studied fluctuations of current in vacuum tubes. [1]

Shot noise may be dominant when the finite number of particles that carry energy (such as electrons in an electronic circuit or photons in an optical device) is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are significant. It is important in electronics, telecommunications, optical detection, and fundamental physics.

The term can also be used to describe any noise source, even if solely mathematical, of similar origin. For instance, particle simulations may produce a certain amount of "noise", where because of the small number of particles simulated, the simulation exhibits undue statistical fluctuations which don't reflect the real-world system. The magnitude of shot noise increases according to the square root of the expected number of events, such as the electric current or intensity of light. But since the strength of the signal itself increases more rapidly, the relative proportion of shot noise decreases and the signal-to-noise ratio (considering only shot noise) increases anyway. Thus shot noise is most frequently observed with small currents or low light intensities that have been amplified.

The number of photons that are collected by a given detector varies, and follows a Poisson distribution, depicted here for averages of 1, 4, and 10. Poisson pmf.svg
The number of photons that are collected by a given detector varies, and follows a Poisson distribution, depicted here for averages of 1, 4, and 10.

Signal-to-Noise

For large numbers, the Poisson distribution approaches a normal distribution about its mean, and the elementary events (photons, electrons, etc.) are no longer individually observed, typically making shot noise in actual observations indistinguishable from true Gaussian noise. Since the standard deviation of shot noise is equal to the square root of the average number of events N, the signal-to-noise ratio (SNR) is given by:

Thus when N is very large, the signal-to-noise ratio is very large as well, and any relative fluctuations in N due to other sources are more likely to dominate over shot noise. However, when the other noise source is at a fixed level, such as thermal noise, or grows slower than , increasing N (the DC current or light level, etc.) can lead to dominance of shot noise.

Properties

Electronic devices

Shot noise in electronic circuits consists of random fluctuations of DC current, which is due to electric current being the flow of discrete charges (electrons). Because the electron has such a tiny charge, however, shot noise is of relative insignificance in many (but not all) cases of electrical conduction. For instance 1 ampere of current consists of about 6.24×1018 electrons per second; even though this number will randomly vary by several billion in any given second, such a fluctuation is minuscule compared to the current itself. In addition, shot noise is often less significant as compared with two other noise sources in electronic circuits, flicker noise and Johnson–Nyquist noise. However, shot noise is temperature and frequency independent, in contrast to Johnson–Nyquist noise, which is proportional to temperature, and flicker noise, with the spectral density decreasing with increasing frequency. Therefore, at high frequencies and low temperatures shot noise may become the dominant source of noise.

With very small currents and considering shorter time scales (thus wider bandwidths) shot noise can be significant. For instance, a microwave circuit operates on time scales of less than a nanosecond and if we were to have a current of 16 nanoamperes that would amount to only 100 electrons passing every nanosecond. According to Poisson statistics the actual number of electrons in any nanosecond would vary by 10 electrons rms, so that one sixth of the time less than 90 electrons would pass a point and one sixth of the time more than 110 electrons would be counted in a nanosecond. Now with this small current viewed on this time scale, the shot noise amounts to 1/10 of the DC current itself.

The result by Schottky, based on the assumption that the statistics of electrons passage is Poissonian, reads [2] for the spectral noise density at the frequency ,

where is the electron charge, and is the average current of the electron stream. The noise spectral power is frequency independent, which means the noise is white. This can be combined with the Landauer formula, which relates the average current with the transmission eigenvalues of the contact through which the current is measured ( labels transport channels). In the simplest case, these transmission eigenvalues can be taken to be energy independent and so the Landauer formula is

where is the applied voltage. This provides for

commonly referred to as the Poisson value of shot noise, . This is a classical result in the sense that it does not take into account that electrons obey Fermi–Dirac statistics. The correct result takes into account the quantum statistics of electrons and reads (at zero temperature)

It was obtained in the 1990s by Viktor Khlus, Gordey Lesovik (independently the single-channel case), and Markus Büttiker (multi-channel case). [2] This noise is white and is always suppressed with respect to the Poisson value. The degree of suppression, , is known as the Fano factor. Noises produced by different transport channels are independent. Fully open () and fully closed () channels produce no noise, since there are no irregularities in the electron stream.

At finite temperature, a closed expression for noise can be written as well. [2] It interpolates between shot noise (zero temperature) and Nyquist-Johnson noise (high temperature).

Examples

  • Tunnel junction is characterized by low transmission in all transport channels, therefore the electron flow is Poissonian, and the Fano factor equals one.
  • Quantum point contact is characterized by an ideal transmission in all open channels, therefore it does not produce any noise, and the Fano factor equals zero. The exception is the step between plateaus, when one of the channels is partially open and produces noise.
  • A metallic diffusive wire has a Fano factor of 1/3 regardless of the geometry and the details of the material. [3]
  • In 2DEG exhibiting fractional quantum Hall effect electric current is carried by quasiparticles moving at the sample edge whose charge is a rational fraction of the electron charge. The first direct measurement of their charge was through the shot noise in the current. [4]

Effects of interactions

While this is the result when the electrons contributing to the current occur completely randomly, unaffected by each other, there are important cases in which these natural fluctuations are largely suppressed due to a charge build up. Take the previous example in which an average of 100 electrons go from point A to point B every nanosecond. During the first half of a nanosecond we would expect 50 electrons to arrive at point B on the average, but in a particular half nanosecond there might well be 60 electrons which arrive there. This will create a more negative electric charge at point B than average, and that extra charge will tend to repel the further flow of electrons from leaving point A during the remaining half nanosecond. Thus the net current integrated over a nanosecond will tend more to stay near its average value of 100 electrons rather than exhibiting the expected fluctuations (10 electrons rms) we calculated. This is the case in ordinary metallic wires and in metal film resistors, where shot noise is almost completely cancelled due to this anti-correlation between the motion of individual electrons, acting on each other through the coulomb force.

However this reduction in shot noise does not apply when the current results from random events at a potential barrier which all the electrons must overcome due to a random excitation, such as by thermal activation. This is the situation in p-n junctions, for instance. [5] [6] A semiconductor diode is thus commonly used as a noise source by passing a particular DC current through it.

In other situations interactions can lead to an enhancement of shot noise, which is the result of a super-poissonian statistics. For example, in a resonant tunneling diode the interplay of electrostatic interaction and of the density of states in the quantum well leads to a strong enhancement of shot noise when the device is biased in the negative differential resistance region of the current-voltage characteristics. [7]

Shot noise is distinct from voltage and current fluctuations expected in thermal equilibrium; this occurs without any applied DC voltage or current flowing. These fluctuations are known as Johnson–Nyquist noise or thermal noise and increase in proportion to the Kelvin temperature of any resistive component. However both are instances of white noise and thus cannot be distinguished simply by observing them even though their origins are quite dissimilar.

Since shot noise is a Poisson process due to the finite charge of an electron, one can compute the root mean square current fluctuations as being of a magnitude [8]

where q is the elementary charge of an electron, Δf is the single-sided bandwidth in hertz over which the noise is considered, and I is the DC current flowing.

For a current of 100 mA, measuring the current noise over a bandwidth of 1 Hz, we obtain

If this noise current is fed through a resistor a noise voltage of

would be generated. Coupling this noise through a capacitor, one could supply a noise power of

to a matched load.

Detectors

The flux signal that is incident on a detector is calculated as follows, in units of photons:

where c is the speed of light, and h is the Planck constant. Following Poisson statistics, the photon noise is calculated as the square root of the signal:

The SNR for a CCD camera can be calculated from the following equation: [9]

where:

Optics

In optics, shot noise describes the fluctuations of the number of photons detected (or simply counted in the abstract) because they occur independently of each other. This is therefore another consequence of discretization, in this case of the energy in the electromagnetic field in terms of photons. In the case of photon detection, the relevant process is the random conversion of photons into photo-electrons for instance, thus leading to a larger effective shot noise level when using a detector with a quantum efficiency below unity. Only in an exotic squeezed coherent state can the number of photons measured per unit time have fluctuations smaller than the square root of the expected number of photons counted in that period of time. Of course there are other mechanisms of noise in optical signals which often dwarf the contribution of shot noise. When these are absent, however, optical detection is said to be "photon noise limited" as only the shot noise (also known as "quantum noise" or "photon noise" in this context) remains.

Shot noise is easily observable in the case of photomultipliers and avalanche photodiodes used in the Geiger mode, where individual photon detections are observed. However the same noise source is present with higher light intensities measured by any photo detector, and is directly measurable when it dominates the noise of the subsequent electronic amplifier. Just as with other forms of shot noise, the fluctuations in a photo-current due to shot noise scale as the square-root of the average intensity:

The shot noise of a coherent optical beam (having no other noise sources) is a fundamental physical phenomenon, reflecting quantum fluctuations in the electromagnetic field. In optical homodyne detection, the shot noise in the photodetector can be attributed to either the zero point fluctuations of the quantised electromagnetic field, or to the discrete nature of the photon absorption process. [10] However, shot noise itself is not a distinctive feature of quantised field and can also be explained through semiclassical theory. What the semiclassical theory does not predict, however, is the squeezing of shot noise. [11] Shot noise also sets a lower bound on the noise introduced by quantum amplifiers which preserve the phase of an optical signal.

See also

Related Research Articles

Noise figure (NF) and noise factor (F) are figures of merit that indicate degradation of the signal-to-noise ratio (SNR) that is caused by components in a signal chain. These figures of merit are used to evaluate the performance of an amplifier or a radio receiver, with lower values indicating better performance.

Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in decibels. A ratio higher than 1:1 indicates more signal than noise.

Specific detectivity, or D*, for a photodetector is a figure of merit used to characterize performance, equal to the reciprocal of noise-equivalent power (NEP), normalized per square root of the sensor's area and frequency bandwidth.

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.

<span class="mw-page-title-main">Johnson–Nyquist noise</span> Electronic noise due to thermal vibration within a conductor

Johnson–Nyquist noise is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise increases with temperature. Some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to reduce thermal noise in their circuits. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.

In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.

An avalanche photodiode (APD) is a highly sensitive semiconductor photodiode detector that exploits the photoelectric effect to convert light into electricity. From a functional standpoint, they can be regarded as the semiconductor analog of photomultiplier tubes. The avalanche photodiode was invented by Japanese engineer Jun-ichi Nishizawa in 1952. However, study of avalanche breakdown, microplasma defects in silicon and germanium and the investigation of optical detection using p-n junctions predate this patent. Typical applications for APDs are laser rangefinders, long-range fiber-optic telecommunication, and quantum sensing for control algorithms. New applications include positron emission tomography and particle physics.

<span class="mw-page-title-main">Photocathode</span>

A photocathode is a surface engineered to convert light (photons) into electrons using the photoelectric effect. Photocathodes are important in accelerator physics where they are utilised in a photoinjector to generate high brightness electron beams. Electron beams generated with photocathodes are commonly used for free electron lasers and for ultrafast electron diffraction. Photocathodes are also commonly used as the negatively charged electrode in a light detection device such as a photomultiplier, phototube and image intensifier.

<span class="mw-page-title-main">Quantum efficiency</span> Property of photosensitive devices

The term quantum efficiency (QE) may apply to incident photon to converted electron (IPCE) ratio of a photosensitive device, or it may refer to the TMR effect of a magnetic tunnel junction.

Photon noise is the randomness in signal associated with photons arriving at a detector. For a simple black body emitting on an absorber, the noise-equivalent power is given by

Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero-point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents.

<span class="mw-page-title-main">Image noise</span> Visible interference in an image

Image noise is random variation of brightness or color information in images, and is usually an aspect of electronic noise. It can be produced by the image sensor and circuitry of a scanner or digital camera. Image noise can also originate in film grain and in the unavoidable shot noise of an ideal photon detector. Image noise is an undesirable by-product of image capture that obscures the desired information. Typically the term “image noise” is used to refer to noise in 2D images, not 3D images.

<span class="mw-page-title-main">Noise (electronics)</span> Random fluctuation in an electrical signal

In electronics, noise is an unwanted disturbance in an electrical signal.

<span class="mw-page-title-main">Image sensor format</span> Shape and size of a digital cameras image sensor

In digital photography, the image sensor format is the shape and size of the image sensor.

<span class="mw-page-title-main">Poisson distribution</span> Discrete probability distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1.

An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:

In signal processing, noise is a general term for unwanted modifications that a signal may suffer during capture, storage, transmission, processing, or conversion.

Quantum excitation is the effect in circular accelerators or storage rings whereby the discreteness of photon emission causes the charged particles to undergo a random walk or diffusion process.

Photon statistics is the theoretical and experimental study of the statistical distributions produced in photon counting experiments, which use photodetectors to analyze the intrinsic statistical nature of photons in a light source. In these experiments, light incident on the photodetector generates photoelectrons and a counter registers electrical pulses generating a statistical distribution of photon counts. Low intensity disparate light sources can be differentiated by the corresponding statistical distributions produced in the detection process.

SU(1,1) interferometry is a technique that uses parametric amplification for splitting and mixing of electromagnetic waves for precise estimation of phase change and achieves the Heisenberg limit of sensitivity with fewer optical elements than conventional interferometric techniques.

References

  1. Schottky, W. (1918). "Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern". Annalen der Physik (in German). 362 (23): 541–567. Bibcode:1918AnP...362..541S. doi:10.1002/andp.19183622304. English translation in: On spontaneous current fluctuations in various electrical conductors
  2. 1 2 3 Blanter, Ya. M.; Büttiker, M. (2000). "Shot noise in mesoscopic conductors". Physics Reports . 336 (1–2). Dordrecht: Elsevier: 1–166. arXiv: cond-mat/9910158 . Bibcode:2000PhR...336....1B. doi:10.1016/S0370-1573(99)00123-4. S2CID   119432033.
  3. Beenakker, C.W.J.; Büttiker, M. (1992). "Suppression of shot noise in metallic diffusive conductors" (PDF). Physical Review B . 46 (3): 1889–1892. Bibcode:1992PhRvB..46.1889B. doi:10.1103/PhysRevB.46.1889. hdl: 1887/1116 . PMID   10003850.
  4. V.J. Goldman, B. Su (1995). "Resonant Tunneling in the Quantum Hall Regime: Measurement of Fractional Charge". Science . 267 (5200): 1010–1012. Bibcode:1995Sci...267.1010G. doi:10.1126/science.267.5200.1010. PMID   17811442. S2CID   45371551. See also Description on the researcher's website Archived 2008-08-28 at the Wayback Machine .
  5. Horowitz, Paul and Winfield Hill, The Art of Electronics, 2nd edition. Cambridge (UK): Cambridge University Press, 1989, pp. 431–2.
  6. "Bryant, James, Analog Dialog, issue 24-3". Archived from the original on 2016-09-29. Retrieved 2008-07-24.
  7. Iannaccone, Giuseppe (1998). "Enhanced Shot Noise in Resonant Tunneling: Theory and Experiment". Physical Review Letters. 80 (5): 1054–1057. arXiv: cond-mat/9709277 . Bibcode:1998PhRvL..80.1054I. doi:10.1103/physrevlett.80.1054. S2CID   52992294.
  8. Thermal and Shot Noise. Appendix C. Retrieved from class notes of Prof. Cristofolinini, University of Parma. Archived on Wayback Machine. [url=https://web.archive.org/web/20181024162550/http://www.fis.unipr.it/~gigi/dida/strumentazione/harvard_noise.pdf]
  9. "Signal-to-Noise Ratio". Teledyne Photometrics. Retrieved 8 March 2022.
  10. Carmichael, H. J. (1987-10-01). "Spectrum of squeezing and photocurrent shot noise: a normally ordered treatment". JOSA B. 4 (10): 1588–1603. Bibcode:1987JOSAB...4.1588C. doi:10.1364/JOSAB.4.001588. ISSN   1520-8540.
  11. Leonard., Mandel (1995). Optical coherence and quantum optics. Wolf, Emil. Cambridge: Cambridge University Press. ISBN   9780521417112. OCLC   855969014.